Mon. Not. R. Astron. Soc. 000, 1–7 (2012)
Printed 14 March 2017
(MN LATEX style file v2.2)
Search for serendipitous TNO occultation in X-rays
arXiv:1211.4650v1 [astro-ph.EP] 20 Nov 2012
Hsiang-Kuang
Chang1,2⋆, Chih-Yuan Liu2,3, and Kuan-Ting Chen2
1
Institute of Astronomy, National Tsing Hua University, Hsinchu 30013, Taiwan
of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan
3 LESIA, Paris Observatory, 92195 Meudon, France
2 Department
Accepted November 19, 2012
ABSTRACT
To study the population properties of small, remote objects beyond Neptune’s orbit
in the outer solar system, of kilometer size or smaller, serendipitous occultation search
is so far the only way. For hectometer-sized Trans-Neptunian Objects (TNOs), optical shadows actually disappear because of diffraction. Observations at shorter wave
lengths are needed. Here we report the effort of TNO occultation search in X-rays using RXTE/PCA data of Sco X-1 taken from June 2007 to October 2011. No definite
TNO occultation events were found in the 334 ks data. We investigate the detection
efficiency dependence on the TNO size to better define the sensible size range of our
approach and suggest upper limits to the TNO size distribution in the size range from
30 m to 300 m. A list of X-ray sources suitable for future larger facilities to observe is
proposed.
Key words: occultations – Kuiper Belt – Solar system: formation – stars: neutron
– X-rays: binaries.
1
INTRODUCTION
Serendipitous Trans-Neptunian Object (TNO) occultation
search is so far the only way to explore properties of
kilometer-sized TNOs or smaller. Such search has been
being conducted mainly in optical bands (Bianco et al.
2010; Bickerton, Kavelaars & Welch 2008; Roques et al.
2003) with few reported detections (Roques et al. 2006;
Schlichting et al. 2009, 2012). In the X-ray band, occultation by smaller objects may be detected because of a smaller
Fresnel scale. Following an earlier discovery of putative occultation events, allegedly caused by 100-meter size TNOs,
in the 1996-2002 X-ray data of Sco X-1 taken by the instrument Proportional Counter Array (PCA) on board Rossi
X-ray Timing Explorer (RXTE) (Chang et al. 2006, 2007),
we conducted new RXTE/PCA observations of Sco X-1 from
June 2007 to Oct 2011 to clarify possible instrumental-effect
contamination in that discovery.
In this paper we report the final result of the effort in
the search for serendipitous TNO occultation in X-rays with
RXTE/PCA, taking into account the issue of detection efficiency. Earlier results based on data taken by Oct 2009 have
been published in Liu et al. (2008) and Chang et al. (2011).
Currently such search is only feasible with RXTE/PCA obseravtion of Sco X-1, because of the large effective area of
PCA and because Sco X-1 is the brightest X-ray source in
the sky (Chang et al. 2006, 2007). Since RXTE was decom⋆
E-mail: [email protected]
c 2012 RAS
missioned in January 2012, new efforts have to await possible
larger instruments in the future.
TNOs larger than deca-kilometer size can be detected
directly. Results of those observations can be found in
Fuentes et al. (2010) and Fuentes, Trilling & Holman (2011)
for works using HST, in Fraser, Brown & Schwamb (2010)
for Subaru, and in Petit et al. (2011) for CFHT. The TNO
size distribution, from thousand-kilometer to decameter size
and smaller, carries infromation of the collisional and dynamical history of the early solar system. It is of essential
importance to our understanding of how the solar system
formed (see e.g. Chiang et al. (2007), Benavidez & Bagatin
(2009), Fraser (2009) and Kenyon & Bromley (2012)).
2
SUMMARY OF THE RXTE/PCA 2007-2011
OBSERVATIONS
RXTE had observed Sco X-1 for many times, and yielded
a large amount of data by 2007 (for details of RXTE instruments, see Jahoda et al. (2006)). Many millisecond-dip
events were found in those data (Chang et al. 2006), but
later it was pointed out that ‘Very Large Events’ (VLEs)
are likely the cause of those dips, rather than TNO occultations (Chang et al. 2007; Jones et al. 2008). VLEs are events
that deposit more than 100 keV into an individual anode in
a Proportional Counter Unit (PCU). RXTE/PCA consists
of 5 identical PCUs. An instrument dead-time of 50 µs is set
for each VLE during RXTE observations of Sco X-1. VLEs
2
Chang, Liu, and Chen
Type
A
B
C
D
number of triggered anodes
average count rate
no
all
more than one but not all
only one
1.73 ± 0.29
34.5 ± 11.0
44.7 ± 9.97
9.37 ± 2.17
Table 1. VLE types and their average count rates (in units of
counts per second per PCU). The number of PCUs which are on
during the observation varies from time to time. The total data
employed in this analysis is 334 ksec, and is 1363 ksec-PCU when
the number of PCU on is taken into account.
Group
associated VLE type
A
B
C
D
E
Type A
Type B, no Type A
Type C, no Type A, B
Type D only
no VLEs
significant
dips
less significant
dips
45
2
3
0
1
180
142
35
6
10
Table 2. Number of dips in different groups. ‘Significant’ dips are
those with the number of counts smaller than −6.5σ below the
average in an 8-second running window, where σ is the standard
deviation of the counts in each bin in the running window. ‘Less
significant’ dips are those below −5.0σ but higher than −6.5σ.
are produced by high-energy photons or particles. Whether
they are really the cause of the millisecond dip events found
earlier, however, was not conclusive because VLEs were only
recorded in housekeeping data with a coarse time resolution
of 125 ms in the data taken by 2007. In order to clarify the
possible instrumental effect that may have caused the millisecond dip events found in Chang et al. (2006, 2007), new
observations with a newly designed data mode to record
more detailed information of each individual VLE, such as
the identification of the triggered PCU and the triggered anode and timing with 125 µs accuracy, were conducted from
June 2007 to October 2011, which produced in total 334 ks
data good for the search for serendipitous TNO occultations.
As already reported in Chang et al. (2011), if one distinguishes VLEs into different types according to the number of anodes that were triggered, one can find that indeed
millisecond dip events are associated with VLEs of different
types in a systematic manner. The average count rate for all
VLEs altogether in the 334-ks data is 90.3 ± 18.8 counts per
second per PCU. The rate for each type of VLEs is shown in
Table 1. To see the association of dip events with VLEs, we
divide dip events into groups according to their associated
VLE types, as shown in Table 2. For more detailed description of these classifications, as well as the algorithm that
we used to find and define the dip event, see Chang et al.
(2011).
From Table 2 we can see that 45 of the 51 significant
dips are of Group A. Type-A VLE only has a count rate of
1.73 ± 0.29 counts per second per PCU. Apparently TypeA VLEs are very strong events that cause these dips, although exact details and why they are recorded with no
anode triggered are not yet understood. The issue whether
Type-B, -C, and -D VLEs can also produce millisecond dips
was discussed in Chang et al. (2011) based on the number
of ‘less significant’ dips. In our search algorithm, we in fact
rounded off the value of count deviations to the first decimal, so all those dips with deviation between −4.95σ and
−6.45σ were counted as ‘less significant’ dips. Since most of
the dips are of 2-ms duration, we consider only the case of
2-ms bins (more precisely speaking, each 2-ms bin in fact
is 8/4096 s = 1.95 ms). There are 1.71 × 108 ’2-ms’ bins in
total in the 334-ks data. The expected number of ‘less significant’ dips due to random fluctuation is 63.4 if a Gaussian
distribution is assumed. The deviation distribution of the
data is a dead-time-corrected Poisson distribution, which is
about a factor of 2 ∼ 3 smaller than a Gaussian at about
−5σ; see the discussion and Figure 1 in Chang et al. (2011).
We therefore consider the total number of less significant
dips to be 63.4/2.5 = 25.4. Among the 1.71 × 108 bins,
9.1 × 107 bins are without the occurrence of any VLEs. The
number of these 25.4 dips in Group E can be estimated to
be 25.4 × 9.1/17.1 = 13.5. That of the other groups can
be estimated with the corresponding VLE count rates. We
therefore expect the numbers of ‘less significant’ dips to be
0.23, 4.54, 5.88, 1.23, and 13.5 for Group A to E respectively. From Table 2 one can see that only that of Group
E is consistent with random fluctuation. The numbers of
‘less significant’ dips of other groups are clearly larger. The
deviation is the largest for Group A, and gets smaller and
smaller from Group A to D. The above estimate is only a
rough one because (1) we consider only the case of 2-ms
bins, i.e. treating all the dips as of duration 2 ms, (2) in
our definition of dip groups, Type B VLEs can also occur
in Group A dips, similarly for Type C and D VLEs, so the
use of each VLE count rates for the estimation is not an
accurate one, and (3) the factor 2.5 to account for the difference from a Gaussian distribution is somewhat arbitrary.
Nonetheless, we think the above estimate is good enough to
indicate the VLE-association of those dips in Group A to D.
Those significant dips are not due to random fluctuation. The above discussion strongly suggests that those of
Group A to D are related to the occurence of VLEs and are
therefore instrumental due to a process not yet fully understood. The ‘significant’ dip event in Group E, denoted as
‘Event E1’ in Chang et al. (2011), is the only non-random
dip event that shows no indication of any instrumental effect. Its light curve was fit with shadow diffraction patterns
in Chang et al. (2011) and the result suggests that it is probably not due to TNO occultation either.
In this section we update the results presented in
Chang et al. (2011) and have the same conclusion: (1) The
fact that most of the ‘significant’ dip events are in Group
A indicates that Group-A dips are instrumental. (2) The
numbers of ‘less significant’ dip events of Group A, B, C
and D are larger than expected from random fluctuation. It
strongly suggests that VLEs, even for Type D, which triggers only one anode, can result in millisecond dip events.
Only those dips without any association with VLEs can be
considered non-instrumental.
3
THE DETECTION EFFICIENCY
In our earlier study to infer the TNO size distribution, a
100% detection efficiency was assumed. To better underc 2012 RAS, MNRAS 000, 1–7
Serendipitous TNO occultation in X-rays
stand the detection efficieny of our approach, we conducted
simulations to investigate the dependence of detection efficiency on the size and relative transverse speed of the occulting body. The detection efficiency surely also depends on the
observed count rate, which is related to the brightness of the
occulted background source and the instrument employed.
In the followng we consider two different count rates. One
is 8 × 104 cps, which is close to what we usually have for
RXTE/PCA onservations of Sco X-1. The other is 2 × 106
cps, which is about 20 times more than RXTE/PCA and
could be achieved by future missions such as the proposed
LOFT project (Mignani et al. 2011). The orbital speed of
the Earth is about 30 km/s. That at 40 AU is 5 km/s. The
orbital speed of the RXTE spacecraft relative to the Earth is
7.8 km/s. Sco X-1 is in a direction at about 5.5◦ to the north
of the ecliptic. The relative transever (to the line of sight)
speed of an occulting body is usually largest when Sco X-1
is in opposition (around end May) and smallest when the
Earth moves towards or away from the direction of Sco X-1.
The orientation of the motion of RXTE and of the occulting
body will also affect that speed. We therefore consider three
different speeds, i.e. 5, 15 and 25 km/s, in the simulation
respectively.
For a given count rate and a given transverse speed,
we would like to know the chance of detecting an occultation event caused by an occulting body of a certain size
with our searching algorithm. Most of the RXTE/PCAobserved photons from Sco X-1 are about 4 keV in energy (0.3 nm
p in wavelength). The Fresnel scale, which is
defined as
λd/2, where λ is the wavelength and d the
distance, is 30 m for λ = 0.3 nm and d = 40 AU. One expects to detect the shadow of an occulting body much larger
than the Fresnel scale easily, while that of a body much
smaller than the Fresnel scale will elude being detected due
to diffraction. Another factor significantly affecting the observed shadow (light curve) is the impact parameter β, usually defined to be the shortest distance of the passing path
to the shadow center in units of the occulting body radius. A
few shadow diffraction patterns, computed with the recipes
in Roques, Moncuquet & Sicardy (1987) and with a typical
RXTE/PCA-observed Sco X-1 photon spectrum, are shown
in Figure 1. We note that with a point background source
the diffraction patterns, expressed in units of the Fresnel
scale, are all the same for a fixed ratio of the occulting body
size to the Fresnel scale. That is, there is a size-distance degeneracy. Attempts to break this degeneracy are discussed
in the next section. We adopt the definition of the shadow
boundary employed in Nihei et al. (2007), in which the relation between the shadow size Ω and the occulting body
√ 3/2
radius ρ is described as Ω = 2( 3
+ ρ3/2 )2/3 , in which
both Ω and ρ are in units of a Fresnel scale. Throughout
this paper, when referring to the occulting body size we will
use the diameter D, which is obviously equal to 2ρ.
In our simulation, with a given count rate F , a given
transverse speed v, and a given occulting body diameter D,
we randomly pick an impact parameter β in the range from
Ω
) with
the shadow center (β = 0) to its boundary (β = D
a uniform probability and a random epoch for each individual implanted event in a 100-s segment. We first compute
such a 100-s model light curve, then bin the light curve into
0.25-ms bins, and then randomize the counts in each bin
c 2012 RAS, MNRAS 000, 1–7
3
Figure 1. Computed shadow light curves for occulting bodies
of different diameter D. All the 4 panels are for the case of a
central-crossing event, that is, for a zero impact parameter β. A
typical RXTE/PCA observed Sco X-1 spectrum is used for the
diffraction pattern computation. Ω is the shadow width as defined
in Nihei et al. (2007). D and Ω are both in units of a Fresnel scale.
Figure 2. The detection efficiency of our algorithm in the search
for serendipitous TNO occultations as a function of the occulting body diameter. Three different relative transverse speeds are
considered. We assume these occulting bodies are at 40 AU away,
that is, one Fresnel scale corresponds to 30 m. The upper panel is
for the case of the typical RXTE/PCA count rate, and the lower
one is for a much higher count rate, which may be achieved by
future missions.
to produce a simulated binned light curve. This light curve
is then processed with exactly the same algorithm that we
apply to real data. We repeat the procedure 105 times for
each set of F , v and D, and record the number of times of
detection, which we use to represent the detection efficiency.
The result of the detection efficiency determination is
shown in Figure 2. One can see that, with the current
RXTE/PCA count rate, a detection efficiency larger than
50% can be achieved for occulting bodies larger than 2 Fresnel scale (diameter), considering the case of relative transeverse speed being 25 km/s, while for the high count rate
case (2 × 106 cps), that occulting body size can be pushed
down to about 0.6 Fresnel scale.
4
Chang, Liu, and Chen
Figure 3. The difference between the detection efficiency times
shadow width (ηΩ) and the occulting body diameter (D) expressed in the form of their ratio as a function occulting body
diameter. For larger occulting bodies the fractional difference is
small. For smaller ones whose detection efficiency is not yet negligible it becomes large since the shadow width approaches a constant value when the size decreases.
comes one order smaller when the object diameter decreases
from 20 Fresnel scale to 2 Fresnel scale. For the current
RXTE/PCA count rate, it actually becomes relatively negligible for diameter smaller than 1 Fresnel scale. Nonetheless,
dn
as a function of D could be very steep, so the contribudD
tion to the event rate from objects of diameter between 1
and 2 Fresnel scale could be considerable. For the high count
rate case, ηΩv becomes relatively negligible when the object
diameter is smaller than 0.4 Fresnel scale.
We estimate the upper limit to the TNO size distribution at the level of setting N = 1 in Equation (1). Although
different models in the literature give different predictions
for the TNO size distribution of sizes smaller than about 100
km, a common trait among most models is a wavy shape,
caused by the competetion of coagulation, erosion and shattering in different size ranges, down to a small size of about
0.1 km (Kenyon & Bromley 2012), 0.5 km (Fraser 2009), or
1 km or so (Benavidez & Bagatin 2009), below which the
collisional equilibrium is achieved. In the collisional equilibrium, one expects to have (O’Brien & Greenberg 2003)
dn
∼ D−q and
dD
q=
7 + p/3
,
2 + p/3
(2)
where p is the power index of the strength-size relation.
In the so-called ‘strength-scaled regime’ (in contrast to
the ‘gravity-scaled regime’ for larger bodies), p is negative.
Most models give a q in the range from 3.5 to 4.0 for the
collisional equilibrium. The distribution can be quite flat
at the small-size end of the wavy-shape distribution. In
Kenyon & Bromley (2012) q is about 1.0 in the range from
0.1 km to 1 km. If we describe the TNO size distribution in
terms of
dn
=
d log D
dn
d log D
D=D0
D
D0
−q+1
,
(3)
we can have, from Equation (1),
Figure 4. The product ηΩv, in units of the Fresnel scale times
km/s, as a function of the occulting body diameter. This product
is a relative contribution to the event rate estimate from TNOs
of different size and different transverse speeds.
For a background point source, the event rate can be
estimated as (cf. Equation (8) in Chang et al. (2011))
N
=
T
R D2
D1
dn
dD
d2
ηΩv dD
×(
180 2
) ,
π
(1)
where N is the number of detected events, T the total expodn
sure time, dD
the differential size distribution (in units of
number per unit length per square degree), η the detection
efficiency, Ω the shadow size, v the typical sky-projection
relative speed, and d the typical distance to the TNOs. The
product ηΩ in the above equation was approximated by the
diameter in Chang et al. (2011) The ratio of them as a function of occulting body diameter D is shown in Figure 3. The
product ηΩv represents a certain relative detection probability. It is plotted in Figure 4 as a function of D for the
three different speeds. We can see that the product ηΩv be-
dn
d log D
D=D0
= RD
2
D1
1
D
D0
−q+1
N d2 π 2
(
) ln 10 , (4)
ηΩ
dD T v 180
D
where v is assumed to be a typical speed independent of D.
We consider two size ranges, that is, object diameter
between 1.0 and 2.0 Fresnel scale and between 2.0 and 10.0
Fresnel scale. Assuming a typical distance d = 40 AU and
a typical relative sky-projection speed v = 25 km/s, with
T = 334 ks and setting one detection as the upper limit,
we have in the size range from 30 m to 60 m (recall that
one Fresnel scale is 30 m in the current case and
whendn
ever we use ’size’ we mean the diameter) d log
<
D
D=30 m
dn
4.0 × 1011 deg−2 for q = 4.0 and d log
< 3.3 ×
D D=30 m
11
−2
10 deg
for q = 3.5. In the size range from 60 m to
dn
300 m we have d log
< 6.6 × 1010 deg−2 for
D
q = 4.0,
dn
d log D D=60 m
D=60 m
< 5.7 × 1010 deg−2 for q = 3.5,
dn
and d log
< 1.2 × 1010 deg−2 for q = 1.0. Another
D D=60 m
factor of 1.14 should be applied to convert the estimate at
the Sco X-1 latitude to the ecliptic; see the discussion right
after Equation (11) in Chang et al. (2011) These upper limits are plotted in Figure 5.
c 2012 RAS, MNRAS 000, 1–7
Serendipitous TNO occultation in X-rays
Sources
4U
4U
4U
4U
4U
4U
4U
4U
4U
4U
1758-25
1758-20
1617-15
1813-14
1702-36
1642-45
1837+04
2142+38
1956+35
Brightness (Crab)
A1
ASMquick
1.21
0.63
17.95
1.00
0.79
0.48
0.30
0.58
1.24
0.56
0.37
9.1
0.33
0.41
0.18
0.14
0.47
0.29
0.58
0.30
11.0
0.37
0.44
0.25
0.12
0.38
0.42
λ
β
269.56
269.66
245.14
273.24
258.11
255.31
280.60
345.98
312.86
-1.63
2.91
5.73
9.35
-13.50
-23.06
28.09
47.96
54.26
5
Other names
1H1758-250, GX5-1
1H1758-205, GX9+1, Sgr X-3
1H1617-155, Sco X-1
1H1813-140, GX17+2, Ser X-2
1H1702-363, GX 349+2, Sco X-2
1H1642-455, GX340+0
1H1837+049, Ser X-1
1H2142+380, Cyg X-2
1H1956+350, Cyg X-1
Table 3. Non-extended bright X-ray sources, which are potential background targets for serendipitous TNO occultation search in
X-rays. All the 9 listed sources are brighter than 0.1 Crab in all the 4th Uhuru (4U), HEAO 1 A-1 (A1), and RXTE All Sky Monitor
(ASMquick) catalogues. The 4U catalog was compiled with observations conducted during 1972-1973, the A1 catalog was during 1977,
and the ASMquick information is based on weekly average in August 2011. Those listed in the 2nd, 3rd and 4th columns are source
brightness in the 4U, A1 and ASMquick catalogues respectively, all in units of a Crab. Their positions are in the 5th and 6th columns,
where λ is the ecliptic longitude and β the ecliptic latitude, both in degrees. This list is sorted with the absolute value of the ecliptic
latitude β.
4
SUMMARY AND DISCUSSION
We conclude the current effort of serendipitous TNO occultation search in X-rays with upper limits to the TNO
size distribution in the size range from 30 m to 300 m. In
RXTE/PCA observations of Sco X-1 from June 2007 to October 2011, only one non-instrumental dip event was found
in the 334-ks data. Due to the distance-size degeneracy in
the occultation light curve, one cannot easily determine the
occulting body size or distance. One way to go around is
to fit the light curve with different diffraction patterns and
find the best-fit result of the relative transverse speed, which
can be obtained only in units of Fresnel scale per unit time.
Then with some orbital assumptions, this speed can be compared to the real relative transverse speed of a body at a
certain distance to find the match (Chang et al. 2011). As
discussed in Chang et al. (2011), based on the fitting result
of the aforementioned approach, that non-instrumental dip
event might be due to a TNO of 150-m size, but with a rare
retrograde orbit, or, it might be due to an MBA of 40-m
size, but the implied size distribution is incredibly steep, or,
it might be due to a very nearby object of meter size moving
at a relative speed of a few kilometers per second. The main
reason of why there exists such a huge uncertainty in the result is the small number of photons detected at millisecond
time scale.
Another possible way to break the distance-size degeneracy is to relax the assumption of the background star being a point source. The size of the X-ray emitting region
in Sco X-1 is still under debate, ranging from about 50 km
to 50,000 km, depending on different models. Sco X-1 is
at a distance about 2.8 kpc. The correspondig angular size
of its X-ray emitting region is 0.1 Fresnel scale at 40 AU
considering the 50,000 km case. It is a good approximation
to assume a point background source in such a case. On
the other hand, if we are confident on the knowledge of the
emitting region size and our instrument is large enough to
provide good photon statistics, with a given angular size of
the background source, one can detect the difference among
diffraction shadow patterns caused by occulting bodies of
the same size in units of Fresnel scale at different distances,
c 2012 RAS, MNRAS 000, 1–7
since the projection size of the background source at different distances are different in units of Fresnel scale. For Sco
X-1, if the occulting body is in the inner Oort Cloud, say,
at 4000 AU, its X-ray emitting region will appear to be 1Fresnel scale large at that distance for the 50,000-km case.
From detailed study of the occultation light curve, one may
be able to distinguish the projection size of the background
source in units of Fresnel scale and therefore to determine
the distance. Again, this requires a larger instrument to provide good enough photon statistics.
After 16 years of fruitful service, RXTE was decommissioned in January 2012. Future projects of building
larger instruments have been proposed, such as Athena
(Barcons et al. 2012), AXTAR (Ray et al. 2011), and LOFT
(Mignani et al. 2011). In particular, LOFT will be 20 times
larger than RXTE/PCA in terms of its photon collecting
power. In Table 3 and 4 we list 9 brightest non-extended
X-ray sources suitable for serendipitous TNO occultation
search in the LOFT era. X-ray sources tend to be variable.
Those 9 sources are all brighter than 0.1 Crab in all the
4th Uhuru, HEAO 1 A1 and RXTE All Sky Monitor catalogues. See table captions for more details. Those 9 sources
are located at different ecliptic latitude, providing the possiblity to study latitude dependence of TNO distribution
in the decameter to hectometer size range. Their estimated
LOFT count rates are also high enough to allow investigation in two or three energy bands, which can help to identify
diffraction features and to determine whether the discovered
dip events are occultation in nature. One of the 9 sources
is a High-Mass X-rar Binary (HMXB) containing a black
hole candidate. The other 8 are all Low-Mass X-ray Binaries (LMXBs), two of which are the so-called ‘Atoll sources’
and the other 6 are ‘Z sources’. They are all neutron-star
systems showing various rich variation in their timing and
spectral properties. Long-term monitoring of these 9 sources
is by itself very much rewarding for the study of accretion
physics around black holes and neutron stars, and is at the
same time useful for pinning down the TNO size distribution
in the decameter and hectometer size range.
6
Chang, Liu, and Chen
Sources
band (keV)
LOFT count rate
count rate (cps)
source fraction
Brightness
(Crab)
Type
4U 1758-25
1-2
2-4
4-6
6-10
10-20
4.8 × 103
1.3 × 105
1.2 × 105
8.3 × 104
2.1 × 104
92.0%
99.7%
99.7%
99.4%
95.4%
1.02
Z source, Cyg X-2 like
4U 1758-20
1-2
2-4
4-6
6-10
10-20
1.3 × 104
6.6 × 104
5.1 × 104
4.1 × 104
1.2 × 104
97.0%
99.5%
99.4%
98.8%
92%
0.48
Atoll source
4U 1617-15
1-2
2-4
4-6
6-10
10-20
1.8 × 106
2.5 × 106
1.1 × 106
8.0 × 105
2.4 × 105
100%
100%
100%
99.9%
99.6%
12.6
Z source (Sco X-1)
4U 1813-14
1-2
2-4
4-6
6-10
10-20
6.9 × 103
6.8 × 104
5.6 × 104
4.1 × 104
1.4 × 104
94.4%
99.5%
99.4%
98.8%
93.1%
0.50
Z source, Sco X-1 like
4U 1702-36
1-2
2-4
4-6
6-10
10-20
1.8 × 104
9.1 × 104
7.1 × 104
5.5 × 104
1.7 × 104
97.8%
99.6%
99.6%
99.1%
94.3%
0.66
Z source, Sco X-1 like
4U 1642-45
1-2
2-4
4-6
6-10
10-20
1.3 × 103
3.8 × 104
4.4 × 104
3.0 × 104
7.3 × 103
69.1%
99.1%
99.3%
98.3%
86.8%
0.34
Z source, Cyg X-2 like
4U 1837+04
1-2
2-4
4-6
6-10
10-20
1.8 × 104
3.3 × 104
2.0 × 104
1.7 × 104
7.1 × 103
97.9%
98.9%
98.4%
97.1%
86.4%
0.21
Atoll source
4U 2142+38
1-2
2-4
4-6
6-10
10-20
1.7 × 104
7.6 × 104
4.6 × 104
3.0 × 104
8.6 × 103
97.8%
99.5%
99.3%
98.4%
88.8%
0.44
Z source (Cyg X-2)
4U 1956+35
1-2
2-4
4-6
6-10
10-20
4.6 × 104
7.3 × 104
4.5 × 104
4.4 × 104
3.0 × 104
99.1%
99.5%
99.3%
98.9%
96.8%
0.5
black hole candidate, HMXB
Table 4. Estimated LOFT count rates for the 9 sources listed in Table 1. These count rates are estimated with LOFT response files
and are based on spectral models reported in the literature (4U 1758-25: Jackson, Church & Balucinska-Church (2009); 4U 1758-20:
Iaria et al. (2005); 4U 1617-15: Bradshaw, Geldzahler & Fomalont (2003); 4U 2142+38: Balucinska-Church et al. (2010); 4U 1956+35:
Nowak et al. (2011); the other 4 sources: Cackett et al. (2009)). Those listed count rates are the total count rate (including background).
The brightness, in units of a Crab, in the 5th column is based on the assumed spectral model and is for the 2-10 keV band, in which a
Crab is 2.4 × 10−8 erg cm−2 sec−1 .
c 2012 RAS, MNRAS 000, 1–7
Serendipitous TNO occultation in X-rays
Figure 5. The TNO size distribution. Plotted here is the differential sky surface density at the ecliptic per decade of size in units
of number per square degree. The upper limits in the size range
of 30-60 m and of 60-300 m, denoted with downward arrows, are
derived from our non-detection in the 334-ks RXTE/PCA data
of Sco X-1, and is set at the level of one detection in 334 ks. The
dn
∼ D −q+1 ) are considered
cases of q equal to 3.5 and 4.0 ( d log
D
in the two size ranges, of which the derived upper limits are very
close to each other as plotted in this figure. The case of q equal to
1.0 is also considered in the size range of 60-300 m. The downward
arrows are plotted at the size-range boundary for q equal to 3.5
and 4.0, and at the middle of the size range for q equal to 1.0. The
asterisk symbol at 0.5 km is based on the reported HST/FGS detection of an occultation event (Schlichting et al. (2009); see also
Schlichting et al. (2012)). The solid curve is the double-power-law
distribution of large TNOs (Fuentes et al. 2010); see Chang et al.
(2011) for the assumptions made to convert the luminosity function to the size distribution. The dashed line is a direct extrapolation from the large size end of the double-power-law towards
smaller size and the dotted line is a power law anchoring on the
double-power-law at 90 km with a power index of -3.0. We note
that the power indices in this figure, explicitly printed close to the
corresponding power-law lines, are for the differential size distribution per decade of size, and are therefore the same as that of
the cumulative size distribution. The double-power-law and the
HST/FGS result in fact indicate a wavy shape for the TNO size
distribution, similar to that of the main-belt asteroids.
ACKNOWLEDGMENTS
We thank Ed Morgan, who designed the new RXTE/PCA
data mode and coordinated the new RXTE observations of
Sco X-1. This work was supported by the National Science
Council of the Republic of China under grants NSC 99-2112M-007 -017 -MY3 and 101-2923-M-007 -001 -MY3.
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